Unable to Upload Package - Access is Denied

We have been unable to upload any new Nuget packages to our Nuget feed. When uploading packages using Nuget.exe, I get a 403-Forbidden: Access is Denied error.

I got a new API key from our feed's owner, but that didn't give a different result.

Is there something we are doing wrong? My only theory now is that we have too many previous versions on the feed. I will have the owner remove older versions and see if that does the trick, but could there be something else that's wrong?